The planning and optimization of wireless communications networks requires accurate propagation models. Propagation predictions are used to estimate quantities such as coverage, serving areas, interference, etc. These quantities, in turn, are used to arrive at equipment settings, such as channel assignments, power levels, antenna orientations, and heights. The goal is to optimize these settings to extract the most capacity and coverage without sacrificing the quality of the network. Thus, it is extremely important to employ a propagation model that is as accurate and reliable as possible. Naturally, the accuracy of the predictions also depends on the quality of the geographical data used as input.
There is another important factor that affects the quality of propagation predictions: the accuracy of the antenna radiation pattern used to estimate the spatial distribution of the transmitted RF power. Accurate pattern information is readily available from antenna manufacturers. Unfortunately, such data are usually available only for cross sections at the vertical and horizontal planes. Since a typical calculation involves arbitrary orientations, a full three-dimensional pattern needs to be generated. It is this step that can introduce considerable error as the spatial distribution of the power radiating from an antenna is generated from only two cross sections. Clearly, the generated pattern will not be unique—the only piece of information that we have is that this surface has to match the two patterns when it intersects the vertical and horizontal planes. In fact, there are an infinite number of 3D surfaces that can be shaped so that they agree with the values available for the two cross sections, and therefore, there is no “correct” generated surface. The best one can hope for is a reasonable estimate and the problem then focuses on finding the algorithm that produces the best estimate.
From a practical point of view, the idea of only using vertical and horizontal data is very attractive, in spite of the uniqueness problem. For example, a pattern stored at one degree increments would require 360×2=720 measured antenna gains. A full 3D surface at the same resolution would require 360×180=64,800 measurements, a number almost two orders of magnitude larger. To our knowledge, no antenna vendor routinely provides this kind of detail. In a limited number of cases, antenna pattern values are available for a few cross sections in addition to the vertical and horizontal. For those cases, one can use the additional information to validate proposed algorithms for 3D surface generation. For most antenna patterns, however, one would still need to rely on some sort of approximation.
As an example of the problem addressed by this invention, a wireless communications link is schematically illustrated in FIG. 1. A typical link includes a transmitting base station 104 and a receiving base station 110 located at some distance from each other. These stations have respective antennas 102 and 108 mounted at some height above the local terrain 114. The quantity of interest is the power that reaches the receiving station 110. This power is given by (in dB units)Pr=Pt+Gt−L+Gr,  (1)where                Pt=transmitter power,        Gt=transmitter antenna gain,        L=propagation path loss,        Gr=receiver antenna gain.        
A popular view is that once the transmitted power and the two antennas 102, 108 are selected, the propagation problem reduces to evaluating the propagation path loss. The path loss is regarded as the difficult part of the calculation and a considerable amount of effort has focused on improving its predictive accuracy. It is interesting to note that even though the literature is full of papers on how to calculate the path loss, not much work on how to apply the antenna patterns has been reported. However, as can be seen in the equation above, errors in the antenna gain terms can be as important as errors in the path loss, especially if the antennas are directional.
FIG. 1 also displays the shape of the antenna patterns 106 and 112 superimposed on the vertical plane defined by the two stations. As can be seen there, these patterns can have a very complex structure with numerous nulls and side lobes. It is also clear that the side lobes can be so sharp that a small error in angle can lead from a peak 116 to a deep null 118, and vice versa.
An antenna pattern is the spatial distribution of the electromagnetic power radiating from an antenna. Typically, the size of the antenna (a couple of meters) is much smaller than the transmitter-receiver distance (a few kilometers) and the antenna can be regarded as a point source. Therefore, it is convenient to analyze a 3D radiation pattern in spherical coordinates, ρ, θ, and φ. In practice, it is desirable to have the θ coordinate defined with respect to the horizontal plane, and therefore, the modified spherical coordinate system shown in FIG. 2 will be used. Here the origin 202 represents the antenna and the point 204 represents some arbitrary location of interest. The radial coordinate 206 represents the antenna gain G, φ is the standard azimuth coordinate and θ represents the angular elevation relative to the X-Y plane. The antenna is mounted on some vertical physical structure oriented along the Z axis. An advantage of this coordinate system is that θ can also be used to describe the amount of electrical tilt applied to the antenna main lobe, which by default will be oriented along the positive X-axis.
A note about terminology: Since the coordinate system used here is similar to the geocentric coordinate system used to describe locations on the surface of the earth and since the unit sphere will be used throughout this paper, it will be convenient to use geographical terminology to describe zones and lines on the surface of the unit sphere. Thus, the equator is defined as the circle, on the X-Y plane, that divides the sphere into northern and southern hemispheres. All points having the same θ form a line called a parallel and all points of the same φ form a meridian line. The prime (φ=0) meridian divides the sphere into east and west hemispheres. Finally, the north and south poles are the points where θ=π/2 and θ=−π/2, respectively. Using this terminology, the horizontal pattern will lie on the equatorial plane and the vertical pattern will lie on the plane defined by the prime meridian.
Notice that the patterns supplied by the antenna manufacturer may not conform to this coordinate system, and indeed, the vertical pattern will not conform and the appropriate coordinate transformation will need to be applied. The reason for this is that the patterns are provided as simple tabulated arrays of gain values. Thus, the vertical array will contain values for vertical angles that usually range from 0 to 2π, while the θ coordinate of FIG. 2 only ranges from −π/2 to π/2. In addition, the direction, clockwise or counterclockwise, will need to be specified.
It is assumed that a set of measured vertical and horizontal patterns, gv(θ′) and gh(φ′) respectively, are available, and that they are normalized to unit maximum gain. These two patterns come from measurements tabulated as functions of vertical and horizontal angles θ′ and φ′, respectively. A schematic (circular) representation of a vertical and horizontal pattern pair is shown in FIGS. 3A and 3B, respectively. These patterns need to be transformed into the coordinate system of FIG. 2. The horizontal pattern 304 is placed on the equatorial plane, while the vertical 302 is placed on the plane of the prime meridian. In order to properly use these values in the coordinate system of FIG. 2, a mapping of the front half of the vertical pattern to the φ=0 meridian and the back half to the φ=π or meridian must be constructed. In other words, the transformation
                              θ          ′                =                  {                                                                      θ                  ,                                                                                                  0                    ≤                    θ                    ≤                                          π                      /                      2                                                        ,                                                                              φ                  =                  0                                                                                                                          π                    -                    θ                                    ,                                                                                                  0                    ≤                    θ                    ≤                                          π                      /                      2                                                        ,                                                                              φ                  =                  π                                                                                                                          π                    -                    θ                                    ,                                                                                                                                                -                        π                                            /                      2                                        ≤                    θ                    <                    0                                    ,                                                                              φ                  =                  π                                                                                                                                                2                      ⁢                      π                                        +                    θ                                    ,                                                                                                                                                -                        π                                            /                      2                                        ≤                    θ                    <                    0                                    ,                                                                              φ                  =                  0                                                                                        Eq        .                                  ⁢                  (          2          )                    is applied when accessing the vertical pattern array. The mapping for the azimuthal coordinate is trivial because φ′ and φ are equivalent, so we write φ=−φ′ or φ=φ′, depending on whether the pattern is tabulated in the clockwise or counterclockwise direction. This coordinate mapping allows the display of the vertical and horizontal pattern pair in 3D, as will be shown below.
There are some important points to make before an antenna radiation pattern in 3D space can be generated. In addition to the uniqueness problem, there is a possible ambiguity about the meaning of the horizontal pattern provided by antenna vendors when the vertical pattern is tilted. To illustrate this, consider the measured vertical 404 and horizontal 406 patterns of FIGS. 4A and 4B. These two patterns correspond to antenna model 1309.17.0007 manufactured by Huber and Suhner. This antenna has an electrical down-tilt of 13 degrees. If the horizontal and vertical patterns are interpreted as cross sections of the 3D pattern surface, the two patterns are expected to coincide at the places where their two planes intersect. In other words,gv(θ′=0)=gh(φ′=0)  Eq. (3)andgv(θ′=π)=gh(φ′=π)  Eq. (4)
It can clearly be seen that in an instance such as FIG. 5, since the main lobe of the vertical pattern 502 lies below the horizontal pattern 504 on the X-Y plane, the maximum vertical gain cannot geometrically match the maximum horizontal gain. Thus, at the outset, inconsistencies for electrically down-tilted antennas are observed. This suggests that the horizontal pattern be shifted up or down until it matches the maximum value of the vertical front lobe. In other words, the horizontal pattern for this case might be considered a shaping function and not a cross section of the 3D surface.
Finally, it is important to recognize that many of the antenna patterns supplied by antenna vendors will not satisfy the above requirements, especially Eq. (4), even when they are not electrically tilted. The inconsistencies may be due to uncertainties in the measured values, or to gaps in the array of measurements. So, in practice, a technique for generating the 3D surface must be robust enough to tolerate inconsistencies at these two points and not produce shape artifacts.